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SINGULARITY OF RANDOM SYMMETRIC MATRICES—A COMBINATORIAL APPROACH TO IMPROVED BOUNDS
Published online by Cambridge University Press: 02 August 2019
Abstract
Let $M_{n}$ denote a random symmetric
$n\times n$ matrix whose upper-diagonal entries are independent and identically distributed Bernoulli random variables (which take values
$1$ and
$-1$ with probability
$1/2$ each). It is widely conjectured that
$M_{n}$ is singular with probability at most
$(2+o(1))^{-n}$. On the other hand, the best known upper bound on the singularity probability of
$M_{n}$, due to Vershynin (2011), is
$2^{-n^{c}}$, for some unspecified small constant
$c>0$. This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of
$M_{n}$ is at most
$2^{-n^{1/4}\sqrt{\log n}/1000}$ for all sufficiently large
$n$. The proof utilizes and extends a novel combinatorial approach to discrete random matrix theory, which has been recently introduced by the authors together with Luh and Samotij.
MSC classification
- Type
- Research Article
- Information
- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Copyright
- © The Author(s) 2019
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